Mean-field theory of strongly disordered superconductors

TL;DR

This paper explores the mean-field theory of strongly disordered superconductors, revealing a smooth BCS-BEC crossover and spatially non-uniform but finite superconducting gaps in disordered systems.

Contribution

It provides a detailed analysis of the mean-field behavior in strongly disordered superconductors, including the effects of disorder and localization on the superconducting gap.

Findings

Exact mean-field solution in the infinite-range limit.

Identification of a smooth BCS-BEC crossover at low densities.

Persistence of a finite superconducting gap below the transition temperature despite disorder.

Abstract

Qualitative features of the mean-field theory of superconductivity in a strongly disordered systems of fermions with short-range attraction are discussed. In this limit the effective theory is entirely bosonic, and I consider both the artificial infinite-range limit, and the more realistic case of "nearest neighbor" hopping of bosons between localized states. In the infinite range case the mean-field theory is exact, and the superconducting gap is uniform in space. There is a smooth BCS-BEC crossover with decrease in density, at weak enough disorder; at moderate densities, or larger disorder, the mean-field ground state is the BCS-like localized superconductor. In the latter case, the gap is highly non-uniform in space, but surprisingly stays everywhere finite below the mean-field transition temperature.